A continuous, constructive solution to Hilbert's \(17^{th}\) problem

DOI10.1007/BF01388465zbMath0547.12017MaRDI QIDQ798716

Charles N. Delzell

Publication date: 1984

Published in: Inventiones Mathematicae (Search for Journal in Brave)

Full work available at URL: https://eudml.org/doc/143128

zbMATH Keywords

sum of squaresreal closureordered fieldsemi-algebraic geometrycontinuous dependence on variables and coefficientscontinuous sums of squarescontinuous, piecewise rational solutionHilbert's \(17^{th}\) problempositive semi-definite polynomialsemi- algebraic setsweighted SOS

Mathematics Subject Classification ID

Forms of degree higher than two (11E76) Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) (12D15) Other constructive mathematics (03F65) Topological fields, rings, etc. (topological aspects) (54H13) Intuitionistic mathematics (03F55) Ordered fields (12J15) General binary quadratic forms (11E16) Forms over real fields (11E10) Real algebraic and real-analytic geometry (14Pxx) Connections of number theory and logic (11U99)

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The Noncommutative Graded Positivstellensatz ⋮ Impossibility of \(C^{\infty}\) variation or formal power series variation in solutions to Hilbert's 17th problem ⋮ Regulous functions ⋮ On mechanical quantifier elimination for elementary algebra and geometry ⋮ A bibliography of quantifier elimination for real closed fields ⋮ European Summer Meeting of the Association for Symbolic Logic (Logic Colloquium '88), Padova, 1988 ⋮ Trilinear equations, Bell polynomials, and resonant solutions ⋮ Continuity properties for flat families of polynomials. I: Continuous parametrizations ⋮ Some continuous Positivstellensätze ⋮ Geometric reasoning with logic and algebra ⋮ Bilinear equations and resonant solutions characterized by Bell polynomials ⋮ Dependence on parameters in analogues of hilberts seventeenth problem ⋮ On the idea(l) of logical closure ⋮ Continuous Pythagoras numbers for rational quadratic forms ⋮ Positivity and Optimization: Beyond Polynomials ⋮ A real Nullstellensatz and Positivstellensatz for the semipolynomials over an ordered field

Cites Work

• Trim stratification of semianalytic sets
• Mathematical significance of consistency proofs
• A finiteness theorem for open semi-algebraic sets, with applications to Hilbert’s 17th problem
• On the Representation of a Rational as a Sum of Four Squares By Means of Regular Functions
• Decision procedures for real and p‐adic fields
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